JEE Main & Advanced Mathematics Sets Laws of Algebra of Sets

(1) Idempotent laws : For any set A, we have

(i)   \[A\cup A=A\]

(ii)  \[A\cap A=A\]

(2) Identity laws : For any set A, we have

(i) \[A\cup \phi =A\]             

(ii) \[A\cap U=A\]

i.e., \[\phi \] and U are identity elements for union and intersection respectively.

(3) Commutative laws : For any two sets A and B, we have

(i) \[A\cup B=B\cup A\]

(ii) \[A\cap B=B\cap A\]

(iii) \[A\Delta B=B\Delta A\]

i.e., union, intersection and symmetric difference of two sets are commutative.

(iv) \[A-B\ne B-A\]

(v) \[A\times B\ne B\times A\]

i.e., difference and cartesian product of two sets are not commutative

(4) Associative laws : If A, B and C are any three sets, then

(i) \[(A\cup B)\cup C=A\cup (B\cup C)\]

(ii) \[A\cap (B\cap C)=(A\cap B)\cap C\]

(iii) \[(A\Delta B)\Delta C=A\Delta (B\Delta C)\]

i.e., union, intersection and symmetric difference of two sets are associative.

(iv) \[(A-B)-C\ne A-(B-C)\]   

(v) \[(A\times B)\times C\ne A\times (B\times C)\]

i.e., difference and cartesian product of two sets are not associative.

(5) Distributive law : If A, B and C are any three sets, then

(i) \[A\cup (B\cap C)=(A\cup B)\cap (A\cup C)\]

(ii) \[A\cap (B\cup C)=(A\cap B)\cup (A\cap C)\]

i.e., union and intersection are distributive over intersection and union respectively.

(iii) \[A\times (B\cap C)=(A\times B)\cap (A\times C)\]

(iv) \[A\times (B\cup C)=(A\times B)\cup (A\times C)\]

(v) \[A\times (B-C)=(A\times B)-(A\times C)\]

(6) De-Morgan’s law : If A, B and C are any three sets, then

(i) \[(A\cup B)'=A'\cap B'\]

(ii) \[(A\cap B)'=A'\cup B'\]

(iii) \[A-(B\cap C)=(A-B)\cup (A-C)\]

(iv) \[A-(B\cup C)=(A-B)\cap (A-C)\]

(7) If A and B are any two sets, then

(i) \[A-B=A\cap B'\]

(ii)  \[B-A=B\cap A'\]

(iii) \[AB=A\Leftrightarrow A\cap B=\phi \]

(iv) \[(AB)\cup B=A\cup B\]

(v) \[(AB)\cap B=\phi \]

(vi) \[A\subseteq B\Leftrightarrow B'\subseteq A'\]

(vii) \[(AB)\cup (BA)=(A\cup B)(A\cap B)\]

(8) If A, B and C are any three sets, then

(i) \[A\cap (BC)=(A\cap B)(A\cap C)\]

(ii) \[A\cap (B\Delta C)=(A\cap B)\Delta (A\cap C)\]